A random variation in the flow of carriers manifests itself as electrical noise. Noise is present in many forms. A common feature among the various forms of noise is that the power associated with these random variations is small. Noise is of particular importance when the electrical network deals with small signals: this is the typical case for radio receivers whose fundamental task is to amplify the information carried by a weak incoming signal without degrading it with its own internally generated noise. The amplification function is typically provided by transistors: the circuit within which the transistor operates defines the microwave network under consideration. Due to the small powers carried by either the signal or the noise, linearity is assumed.
The noise parameters are 4 real numbers that are used to fully characterize the noise performance of a field effect transistor (FET) serving as a 2-port microwave network at a given frequency of operation. Noise parameters are of particular importance to characterize and compare transistors since they are at the core of the design of microwave amplifiers and other networks for low noise applications. Therefore, facilitating the determination of the noise parameters from measurement will benefit both transistor manufacturers and circuit designers for microwave applications.
The set of noise parameters is not unique, since it depends on the type of representation in use to describe the FET. For instance, the noise parameters in Z or T (ABCD) representations are different, yet equivalent since they represent the same device. It is common practice in both industry and research institutions to represent the noise parameters with the minimum noise figure Fmin, usually reported in dB; the real and imaginary part of the optimum noise reflection coefficient ΓSopt; and the equivalent noise resistance Rn.
A FET processing a small signal can be described in the frequency domain by a linear relationship between input and output current (I1; I2) and voltages (V1; V2), or a linear combination of these quantities. See H. Hillbrand and P. H. Russer, “An efficient method for computer aided noise analysis of linear amplifier networks,” IEEE Trans. on Circuits and Systems, Vol. CAS-23, No. 4, pp. 235-238, April 1976. For instance, the T or (ABCD) representation can be described by
                              [                                                                      V                  1                                                                                                      I                  1                                                              ]                =                                                            T                *                                  [                                                                                                              V                          1                                                                                                                                                              -                                                      I                            2                                                                                                                                ]                                            +                              [                                                                                                    υ                        n                                                                                                                                                i                        n                                                                                            ]                                      ↔            y                    =                                    T              *              x                        +                          n              T                                                          (        1        )            where the minus sign refers to the fact that currents are conventionally considered flowing into the device, T is the device's transmission matrix, and x is the vector
      [                                        V            1                                                            -                          I              2                                            ]    ,and where the voltage and current noise sources νn and in, respectively, of the device in that representation are arranged in a vector nT in order to make use of matrix algebra.
The vector nT and its hermitian (i.e. conjugate transpose) nT+ define the correlation matrix CT:
                                          C            T                    =                                                    n                T                            ⁢                              n                T                +                                      =                          4              ⁢                                                          ⁢                                                N                  i                                ⁡                                  [                                                                                                              R                          n                                                                                                                      ρ                          0                          *                                                                                                                                                              ρ                          0                                                                                                                      g                          n                                                                                                      ]                                                                    ,                            (        2        )            where + is the hermitian operation, Ni=kT0B where B is the bandwidth within which Ni is considered, Rn is the equivalent noise resistance, and gn is the equivalent noise conductance. The terms in Equation (2) are all defined in the frequency domain, with the diagonal terms representing the average values of the network's respective noise sources and the off-diagonal terms referring to the correlation between the noise sources.
This correlation matrix uniquely describes the noise performance of the network, and its elements are the noise parameters of the network. These elements can be linked to a measurable quantity known as the “noise figure” F, which measures the deterioration of the signal-to-noise ratio as a signal flows through the noisy network. See H. T. Friis, “Noise Figures of Radio Receivers,” Proc. IRE, Vol. 32, pp. 419-422, July 1944. The noise figure is a dimensionless quantity, usually reported in dB, and is defined within a narrow bandwidth B around the frequency of operation f0 and is dependent on the source admittance YS=GS+jBS, i.e., the output admittance of the equivalent small signal network representation of the source at f0.
Hillibrand, supra, provides an expression of the form
                                          F            ⁡                          (                              Y                s                            )                                =                      1            +                                                            y                  s                  +                                ⁢                                  C                                      T                    meas                                                  ⁢                                  y                  s                                                            4                ⁢                                                                  ⁢                                  N                  i                                ⁢                                  G                  s                                                                    ⁢                                  ⁢        where                            (                  3          ⁢                                          ⁢          a                )                                          y          s          +                =                  [                                                                      Y                  s                                                            1                                              ]                                    (                  3          ⁢                                          ⁢          b                )            to evaluate the noise figure of a noisy network as a function of the source admittance YS. Using mathematical operations known in the art, Equations (3a) and (3b) can be transformed into the standard expression for noise figure:
                              F          ⁡                      (                          Y              S                        )                          =                              F            min                    +                                    R              n                        ⁢                                                                                                                        Y                      S                                        -                                          Y                      Sopt                                                                                        2                                            G                s                                                                        (        4        )            where GS is the conductance of the known source admittance YS, i.e., real(YS)=GS, Fmin is the minimum noise figure, Rn is the equivalent noise resistance, and the complex value YSopt is the source's admittance at the frequency f0. Fmin, Rn and YSopt are the noise parameters.
Equation (4) is the basis of the standard measurement of the noise parameters of a two-port microwave network, and highlights the dependence of the noise figure F on the noise source admittance YS. For example, if YS=(1/50[Ω])=0.02 [S], then Equation (4) produces the noise figure F(YS)=F50 well known in the art. Alternatively, if the noise figure F(YS) is measured for at least four different source admittance values YS, Equation (4) can be solved for Fmin, Rn, GSopt and BSopt where YSopt=GSopt+jBSopt. A least squares method has become the standard method for extracting the noise parameters from the measurement of noise figure versus source admittance where more than 4 measurements are made. See R. Q. Lane, “The determination of device noise parameters,” Proc. IEEE, Vol. 57, pp. 1461-1462, August 1969.
A major deficiency of this standard method, however, is that the measurement setup must be able to present four or more distinct source admittance values YS to the device under test (DUT). This key requirement calls for the use of a microwave tuner, which imposes a constraint on the usable frequency range of the setup.
In addition, the measurement uncertainties associated with this procedure often generates noise parameter values that may not be acceptable, and it is not possible to decide a priori whether a given set of measured pairs (F, YS) will generate acceptable noise parameters. It has been demonstrated that
                                                        4              ⁢                                                          ⁢                              NT                0                                                    T              min                                ≥          1                ,                            (        5        )            where N is the Lange parameter and Tmin=T0*(F−1) is the minimum noise temperature. This inequality is a validity test that can be used during measurement sessions or when reviewing data sheets of commercially available devices and must be verified for the noise parameters presented therein to be acceptable. See L. Boglione, “An original demonstration of the Tmin/T0≦4N inequality for noisy two-port networks,” IEEE Microwave and Wireless Components Letters, Vol. 18, No. 5, pp. 326-328, May 2008.
An alternative method for determining the noise parameters was developed by P. J. Tasker et al. See P. J. Tasker, W. Reinert, B. Hughes, J. Braunstein, and M. Schlechtweg, “Transistor noise parameter extraction using a 50 ohm measurement system,” IEEE MTT S International Microwave Symposium Digest, pp. 1251-1254, Atlanta, 1993. The Tasker method is based on the Pospieszalski model, a simple and popular noise model which characterizes the noise performance of a network through evaluation of its “equivalent noise temperatures.” The idea behind the use of equivalent noise temperatures is the concept that a noise source can be associated with an equivalent temperature through the standard relationship with power: P=kTB. The power P in a frequency bandwidth B is equivalent to the temperature T. The Pospieszalski model makes use of two resistors, RGS and RDS, which generate uncorrelated noise power; their respective noise powers are represented by their equivalent noise temperatures Tgs and Tds. The Pospieszalski noise model has been validated with many types of device, both on-wafer and packaged. See M. W. Pospieszalski, “Modelling of noise parameters of MESFET's and MODFET's and their frequency and temperature dependence,” IEEE Trans. on Microwave Theory and Techniques, Vol. 37, No. 9, pp. 1340-1350, September 1989; L. Boglione, R. D. Pollard, and V. Postoyalko, “The Pospieszalski noise model and the imaginary part of the optimum noise source impedance of extrinsic or packaged FETs,” IEEE Microwave And Guided Wave Letters, Vol. 7, No. 9, pp. 270-272, September 1997; and M. W. Pospieszalski, “Interpreting transistor noise,” Microwave Magazine, IEEE, Vol. 11, No. 6, pp. 61-69, October 2010; see also B. Hughes, “A temperature noise model for extrinsic FET's,” IEEE Trans. on Microwave Theory and Techniques, Vol. 40, No. 9, pp. 1821-1831, September 1992.
The Tasker approach takes advantage of the experimental observation that Tgs, the noise temperature of the Pospieszalski noise model between the source and the gate, approaches the expected value T0≈290K at room temperature. The standard value YS=1/50[Ω]=20[mS] is used during noise figure measurement since microwave instrumentation is generally matched to 50[Ω]. Therefore, one measurement of the noise figure for a single source admittance YS at the operation frequency is sufficient to determine the single unknown Td, because the noise sources represented by Tgs and Tds are uncorrelated according to the Pospieszalski model. The removal of the source tuner from the setup has raised interest in the community. However, in the Tasker method, the noise temperature Tds is not obtained analytically from the measurement data, but instead is obtained by an optimization procedure based on the use of a standard circuit simulator software in which the software user determines the equivalent noise temperature Tds by varying its value, simulating the noise figure of the transistor model, and comparing it against the measured noise figure. In practice, the noise temperature Tgs is also varied around the expected value T0 in order to optimize the simulated noise figure against its measurement.